Theorem ontric3g 43480

Description: For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)

Assertion

Ref	Expression
ontric3g	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ontric3g

Step	Hyp	Ref	Expression
1		orcom 870	. . . . . . 7 ⊢ ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥))
2	1	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
3		onsseleq 6406	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥)))
4		ontri1 6399	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
5	2, 3, 4	3bitr2d 307	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥) ↔ ¬ 𝑥 ∈ 𝑦))
6	5	con2bid 354	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
7	6	ancoms 458	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)))
8	4	ancoms 458	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
9		ontri1 6399	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ ¬ 𝑦 ∈ 𝑥))
10	8, 9	anbi12d 632	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥)))
11		eqss 3981	. . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑦))
12		ioran 985	. . . 4 ⊢ (¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (¬ 𝑥 ∈ 𝑦 ∧ ¬ 𝑦 ∈ 𝑥))
13	10, 11, 12	3bitr4g 314	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)))
14		equcom 2016	. . . . . . 7 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
15	14	orbi2i 912	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦))
16	15	a1i 11	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
17		onsseleq 6406	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ⊆ 𝑦 ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦)))
18	16, 17, 9	3bitr2d 307	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥) ↔ ¬ 𝑦 ∈ 𝑥))
19	18	con2bid 354	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))
20	7, 13, 19	3jca 1128	. 2 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥))))
21	20	rgen2 3186	1 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   ∈ wcel 2107  ∀wral 3050   ⊆ wss 3933  Oncon0 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-tr 5242  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6368  df-on 6369

This theorem is referenced by: (None)